# Learning in the Null Space: Small Singular Values for Continual Learning

arXiv: https://arxiv.org/abs/2602.21919 · HTML: https://mlsanity.com/paper/2602.21919/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 28 references that are in the map.
- First posted on arXiv 25 Feb 2026.

## Builds on (most cited first, 24 of 24)

- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [iCaRL: Incremental Classifier and Representation Learning](https://mlsanity.com/paper/1611.07725/) (2016)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Continual Learning with Deep Generative Replay](https://mlsanity.com/paper/1705.08690/) (2017)
- [Experience Replay for Continual Learning](https://mlsanity.com/paper/1811.11682/) (2018)
- [Efficient Lifelong Learning with A-GEM](https://mlsanity.com/paper/1812.00420/) (2018)
- [Overcoming catastrophic forgetting with hard attention to the task](https://mlsanity.com/paper/1801.01423/) (2018)
- [Orthogonal Gradient Descent for Continual Learning](https://mlsanity.com/paper/1910.07104/) (2019)
- [Gradient Projection Memory for Continual Learning](https://mlsanity.com/paper/2103.09762/) (2021)
- [How Does Mixup Help With Robustness and Generalization?](https://mlsanity.com/paper/2010.04819/) (2020)
- [Continual Learning of Context-dependent Processing in Neural Networks](https://mlsanity.com/paper/1810.01256/) (2018)
- [Adversarial Continual Learning](https://mlsanity.com/paper/2003.09553/) (2020)
- [TRGP: Trust Region Gradient Projection for Continual Learning](https://mlsanity.com/paper/2202.02931/) (2022)
- [Continual Learning and Catastrophic Forgetting](https://mlsanity.com/paper/2403.05175/) (2024)
- [Flattening Sharpness for Dynamic Gradient Projection Memory Benefits Continual Learning](https://mlsanity.com/paper/2110.04593/) (2021)
- [A Comprehensive Survey of Forgetting in Deep Learning Beyond Continual Learning](https://mlsanity.com/paper/2307.09218/) (2023)
- [Continual Learning with Scaled Gradient Projection](https://mlsanity.com/paper/2302.01386/) (2023)
- [Parameter-Efficient Continual Fine-Tuning: A Survey](https://mlsanity.com/paper/2504.13822/) (2025)
- [CODE-CL: Conceptor-Based Gradient Projection for Deep Continual Learning](https://mlsanity.com/paper/2411.15235/) (2024)
- [Pay Attention to Small Weights](https://mlsanity.com/paper/2506.21374/) (2025)
- [FlexRank: Nested Low-Rank Knowledge Decomposition for Adaptive Model Deployment](https://mlsanity.com/paper/2602.02680/) (2026)

## Built on by (most cited first, 1 of 1)

- [Forget to Improve: On-Device LLM-Agent Continual Learning via Budget-Curated Memory](https://mlsanity.com/paper/2606.25115/) (2026)

## Abstract (arXiv)

Alleviating catastrophic forgetting while enabling further learning is a primary challenge in continual learning (CL). Orthogonal-based training methods have gained attention for their efficiency and strong theoretical properties, and many existing approaches enforce orthogonality through gradient projection. In this paper, we revisit orthogonality and exploit the fact that small singular values correspond to directions that are nearly orthogonal to the input space of previous tasks. Building on this principle, we introduce NESS (Null-space Estimated from Small Singular values), a CL method that applies orthogonality directly in the weight space rather than through gradient manipulation. Specifically, NESS constructs an approximate null space using the smallest singular values of each layer's input representation and parameterizes task-specific updates via a compact low-rank adaptation (LoRA-style) formulation constrained to this subspace. The subspace basis is fixed to preserve the null-space constraint, and only a single trainable matrix is learned for each task. This design ensures that the resulting updates remain approximately in the null space of previous inputs while enabling adaptation to new tasks. Our theoretical analysis and experiments on three benchmark datasets demonstrate competitive performance, low forgetting, and stable accuracy across tasks, highlighting the role of small singular values in continual learning. The code is available at https://github.com/pacman-ctm/NESS.